Informative line

Greatest Common Factor (GCF) of  Prime and Co-Prime Numbers

GCF of two or more prime numbers

  • Two prime numbers do not have any common factor other than \(1.\) One \((1)\) is the only common factor of all the prime numbers.
  • GCF of two prime numbers is the highest common factor between both  the prime numbers.
  • So, GCF of two prime numbers is always 1.

For example: Find the GCF of the two prime numbers, \(13\) and \(17\).

Factors of \(13=1×13\)

Factors of \(17=1×17\)

Only 1 is the common factor of \(13\) and \(17\).

Thus, GCF of \(13\) and \(17\) is \(1.\)

GCF of co-prime numbers

  • The co-prime numbers are the set of two or more numbers which has only one (1) as the common factor.
  • Thus, GCF of co-prime numbers is always one (1).

For example:

Find the GCF of the co-prime numbers, 14 and 15.

Factors of 14 = 1 × 2 × 7

Factors of 15 = 1 × 3 × 5

Only 1 is the common factor of 14 and 15.

Thus, GCF of 14 and 15 is 1.

Illustration Questions

Find the GCF of the co-prime numbers, 8 and 9.

A 72

B 64

C 81

D 1

×

Given co-prime numbers: 8 and 9

The co-prime numbers have only one (1) as the common factor.

Factors of 8 = 1 × 2  × 2 × 2

Factors of 9 = 1 × 3  × 3

So, GCF = 1

Hence, option (D) is correct.

Find the GCF of the co-prime numbers, 8 and 9.

A

72

.

B

64

C

81

D

1

Option D is Correct

Greatest Common Factor (GCF)

  • GCF or greatest common factor is also known as the greatest common divisor (GCD).
  • GCF of two numbers is the largest factor which is common in those numbers.
  • We can understand GCF with the help of distributive property as discussed below.

\((24+60)=12(2+5)\)

  • Here, \(12\) is the greatest common factor (GCF) of \(24\) and \(60\).
  • \(2\) and \(5\) are relatively prime or co-prime numbers.

Method of finding GCF

  • Consider the two whole numbers, \(18\) and \(24\).

Factors of  18 = 2, 3, 6, 9

Factors of  24 = 2, 3, 4, 6, 8, 12

  • Here, \(6\) is the greatest common factor of both the whole numbers, \(18\) and \(24\).
  • Through distributive property, we can write them as shown.

  • If we take \(3\) as the GCF of \(18\) and \(24\) then through distributive property, it can be written as: \((18+24)=3(6+8)\)
  • Here, \(6\) and \(8\) are not relatively prime because they have the number \(2\) as the common factor.

Thus, \(3\) is not the GCF of \(18\) and \(24\).

Illustration Questions

Find the GCF of \(27\) and \(63\) for the following expression: \((27+63)\)

A \(3\)

B \(9\)

C \(7\)

D \(18\)

×

Through distributive property, we can write \(27\) and \(63\) as:

image

\(3\) and \(7\) are relatively prime.

\(\therefore\;9\) is the GCF of \(27\) and \(63\).

Hence, option (B) is correct.

Find the GCF of \(27\) and \(63\) for the following expression: \((27+63)\)

A

\(3\)

.

B

\(9\)

C

\(7\)

D

\(18\)

Option B is Correct

GCF by Prime Factorization

  • GCF or greatest common factor is also known as the greatest common divisor (GCD).
  • GCF of two numbers is the largest factor which is common in those numbers.
  • To find the GCF of the given numbers, we should follow the following steps:
  1. First, factorize them with the help of a factor tree to find out their prime factors.
  2. Choose the common prime factors.
  3. Multiply those common factors together and we get the GCF of the given numbers.

For example:

  • Consider the two numbers, \(30\) and \(36\).

Step-1 Find the prime factors of \(30\) and \(36\) with the help of the factor trees as shown.

Prime factors of \(30=2,\;3\) and \(5\)

Prime factors of \(36=2,\;2,\;3\) and \(3\)

Step-2 The common prime factors \(=2\) and \(3\)

Step-3 G.C.F. \(=2×3=6\)

Thus, the GCF of \(30\) and \(36=6\)

Prime factors of \(30=2,\;3\) and \(5\)

Prime factors of \(36=2,\;2,\;3\) and \(3\)

Step-2 The common prime factors \(=2\) and \(3\)

Step-3 G.C.F. \(=2×3=6\)

Thus, the GCF of \(30\) and \(36=6\)

Illustration Questions

Find the GCF of \(45\) and ​\(54\).  

A \(9\)

B \(15\)

C \(6\)

D \(27\)

×

The prime factorization of \(45\) and \(54\) is shown.

image

The common prime factors \(=3\) and \(3\)

G.C.F. \(=\) Product of common prime factors

\(=3×3\)

\(=9\)

Thus, the GCF of \(45\) and \(54=9\)

Hence, option (A) is correct.

Find the GCF of \(45\) and ​\(54\).  

A

\(9\)

.

B

\(15\)

C

\(6\)

D

\(27\)

Option A is Correct

Word Problems of GCF

  • We know how to find the GCF but we do not know how to use its concept to solve the real life problems.
  • To solve the real world problems, we should know what exactly is being asked in the problem.

The problems on GCF may ask us to:

  • Split the things into smaller sections/parts.
  • Equally distribute two or more items into their largest groups.
  • Arrange something into rows, columns or groups.

For example:

Mr. Thompson has \(40\) chocolates and \(60\) candies. Find the largest number of students among whom she can distribute these items evenly.

  • In this example, the items are to be distributed equally among the largest number of students, so we have to find the GCF of the two numbers.

Illustration Questions

Julie has two pieces of cloth. One piece of cloth is \(72\) inches wide and another piece of cloth is \(90\) inches wide. Julie wants to cut both the pieces into ribbons of equal width that should be as wide as possible. How wide should she cut the strips?

A \(18\;\text{inches}\)

B \(16\;\text{inches}\)

C \(17\;\text{inches}\)

D \(20\;\text{inches}\)

×

Julie wants to cut both the pieces into ribbons of equal width. It means, split the things into smaller sections. So, we have to find the GCF of \(72\) and \(90\).

For this, prime factorize \(72\) and \(90\) with the help of the factor trees.

image

The common factors of \(72\) and \(90=2,\;3\) and \(3\)

G.C.F. \(=\) Product of common prime factors

\(=2×3×3\)

\(=18\)

Thus, Julie should cut both the pieces into ribbons of width \(18\) inches.

Hence, option (A) is correct.

Julie has two pieces of cloth. One piece of cloth is \(72\) inches wide and another piece of cloth is \(90\) inches wide. Julie wants to cut both the pieces into ribbons of equal width that should be as wide as possible. How wide should she cut the strips?

A

\(18\;\text{inches}\)

.

B

\(16\;\text{inches}\)

C

\(17\;\text{inches}\)

D

\(20\;\text{inches}\)

Option A is Correct

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